Distribution (Continuous)

continuous uniform distribution: A continuous random variable whose probability distribution is the uniform distribution is often called a uniform random variable. If we know nothing about a random variable apart from the fact that it has a lower and an upper bound, then a uniform distribution is a natural model

uniform distribution plot
uniform distribution plot

uniform distribution content
uniform distribution content

  • mean: $\frac{u+l}{2}$
  • variance: $\frac{(u-l)^2}{12}$

exponential distribution: We assume that failures form a Poisson process in time; then the time to the next failure is exponentially distributed.

exponential distribution plot
exponential distribution plot

exponential distribution formula
exponential distribution formula

  • mean: $\frac{1}{\lambda}$
  • variance: $\frac{1}{\lambda ^ 2}$

Normal Distribution

a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean

normal distribution plot
normal distribution plot

$$ p(x)=(\frac{1}{\sqrt{2\pi}\sigma})\text{exp}(\frac{-(x-\mu)^2}{2\sigma^2}) $$

  • mean: $\mu$
  • variance: $\sigma ^2$
  • 68% data within $\sigma$, 95% data within $2\sigma$, 99% data within $3\sigma$
  • A continuous random variable is a normal random variable if its probability density function is a normal distribution.
  • another name for normal distribution is Gaussian distributions.
  • central limit theorem (CLT): under some not very worrying technical conditions, the sum of a large number of independent random variables will be very close to normal.

standard normal distribution:

$$ p(x)=(\frac{1}{\sqrt{2\pi}})\text{exp}(\frac{-x^2}{2}) $$

  • mean: $0$
  • variance: $1$
  • A continuous random variable is a standard normal random variable if its probability density function is a standard normal distribution.
  • Any probability density function that is a standard normal distribution in standard coordinates is a normal distribution.

binomial distribution approximation:

when $N$ is huge, we can approximate binomial distribution with normal distribution to reduce calculation cost.

Assume h follows the binomial distribution with parameters p and q. Write:

$$ x=\frac{h-Np}{\sqrt{Npq}} $$

The, for large N, the probability distribution $P(x)$ can be approximated by the probability density function:

$$ P(\{x\in [a,b]\})\approx \int^b_a(\frac{1}{\sqrt{2\pi}})\text{exp}(\frac{-x^2}{2}) $$

Experiment

population and sample: if we could have seen everything, is the population. I will write populations like random variables with capital letters to emphasize we don’t actually know the whole population. The data we actually have is the sample.

sample mean: the mean from sample, usually notated as: $X^{(N)}$: sample mean when sample size is $N$

  • it is a random variable
  • $\mathbb{E}[X^{(N)}]=\text{popmean(\{X\})}$
  • $var[X^{(N)}]=\frac{\text{popsd(\{X\})}^2}{N}$
  • $std[X^{(N)}]=\frac{\text{popsd(\{X\})}}{\sqrt{N}}$

confident interval:

  • confidence interval for a population mean: Choose some fraction f, An f confidence interval for
    a population mean is an interval constructed using the sample mean. It has the property that for that fraction f of all samples, the population mean will lie inside the interval constructed from each sample’s mean.
  • centered confidence interval for a population mean: Choose some$0<\alpha<0.5$. A $1-2\alpha$ centered confidence interval for a population mean is an interval $[a,b]$; b constructed using the sample mean.

unbiased standard deviation: use to estimate the population standard deviation

$$ \text{stdunbiased(\{x\})}=\sqrt{\frac{\sum_i(x_i-\text{mean}(\{x\})^2)}{N-1}} $$

standard error: The standard deviation of the estimate of the mean

$$ \text{stderr}(\{x\})=\frac{\text{stdunbiased(\{x\})}}{\sqrt{N}} $$

Random variable distribution

we learn two distribution for random variable for far: the t-distribution and normal distribution, depended on the sample size, if $n<30$, we use t-distribution, otherwise use normal distribution

t-distribution:

$$ T=\frac{\text{mean}(\{x\})-\text{popmean(\{X\})}}{\text{stderr(\{x\})}} $$

degree of freedom: $N-1$

normal distribution:

$$ Z=\frac{\text{mean}(\{x\})-\text{popmean(\{X\})}}{\text{stderr(\{x\})}} $$

degree of freedom: $N$

Practice Question

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